An azimuthly symmetric conical beam is reflected by an interface. Under the assumption of a plane-layered earth, the amplitude at any point of the ring-shaped wavefront depends only on the Snell parameter p and the offset x.
Let's first apply equation (2) for the downgoing beam, at a unit distance from the source, and just before it hits the reflector:
Applying equation (2) for the upcoming beam; just after the reflection and just before it hits the surface, we find that
I define now an average amplitude square over a wavefront, in such a way that it keeps the ``average sign of the wavefront"
It is possible then to have an estimation of the average reflection coefficient over the region illuminated by the beam through the following relation
Dividing equation (4) by equation (3) and using equations (6) and (5) results in
Vertical cross section of the upcoming beam. If we approximate the surface S by a conical surface, the area element ds is given by .
A last necessary assumption is that
Finally, if the wavefront ring reaching the surface is approximated by a conical surface, the application of straightforward geometry leads to
Using this result and substituting equation (8) into equation (7), a final expression for the output sample is found:
All the cosine factors in equation (10) can be expressed in terms of pt or pb, the horizontal slownesses of the boundary rays of the beam, according to
The offsets xt and xb that appear in the integral limits of equation (9) are determined by the interceptions (in the x-t domain) of the top and bottom rays with the reflection curve (or wavefront) t=t(x,t0). The specific function that describes the wavefronts in the x-t domain can be given either explicitly by
if an hyperbolic approximation is satisfactory, or implicitly with the use of raytracing.